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Properties

Label 300.5.g.a.101.1
Level $300$
Weight $5$
Character 300.101
Self dual yes
Analytic conductor $31.011$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,5,Mod(101,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.101");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 300.g (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.0109889252\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 101.1
Character \(\chi\) \(=\) 300.101

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} -71.0000 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q-9.00000 q^{3} -71.0000 q^{7} +81.0000 q^{9} -191.000 q^{13} -601.000 q^{19} +639.000 q^{21} -729.000 q^{27} +1559.00 q^{31} +2062.00 q^{37} +1719.00 q^{39} -3191.00 q^{43} +2640.00 q^{49} +5409.00 q^{57} +7199.00 q^{61} -5751.00 q^{63} +8809.00 q^{67} +8542.00 q^{73} +7682.00 q^{79} +6561.00 q^{81} +13561.0 q^{91} -14031.0 q^{93} -9071.00 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −9.00000 −1.00000
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −71.0000 −1.44898 −0.724490 − 0.689286i \(-0.757925\pi\)
−0.724490 + 0.689286i \(0.757925\pi\)
\(8\) 0 0
\(9\) 81.0000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −191.000 −1.13018 −0.565089 − 0.825030i \(-0.691158\pi\)
−0.565089 + 0.825030i \(0.691158\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −601.000 −1.66482 −0.832410 − 0.554160i \(-0.813039\pi\)
−0.832410 + 0.554160i \(0.813039\pi\)
\(20\) 0 0
\(21\) 639.000 1.44898
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −729.000 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1559.00 1.62227 0.811134 − 0.584860i \(-0.198851\pi\)
0.811134 + 0.584860i \(0.198851\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2062.00 1.50621 0.753104 − 0.657901i \(-0.228555\pi\)
0.753104 + 0.657901i \(0.228555\pi\)
\(38\) 0 0
\(39\) 1719.00 1.13018
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −3191.00 −1.72580 −0.862899 − 0.505377i \(-0.831354\pi\)
−0.862899 + 0.505377i \(0.831354\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 2640.00 1.09954
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5409.00 1.66482
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 7199.00 1.93469 0.967347 − 0.253454i \(-0.0815666\pi\)
0.967347 + 0.253454i \(0.0815666\pi\)
\(62\) 0 0
\(63\) −5751.00 −1.44898
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8809.00 1.96235 0.981176 − 0.193115i \(-0.0618589\pi\)
0.981176 + 0.193115i \(0.0618589\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 8542.00 1.60293 0.801464 − 0.598043i \(-0.204055\pi\)
0.801464 + 0.598043i \(0.204055\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7682.00 1.23089 0.615446 − 0.788179i \(-0.288976\pi\)
0.615446 + 0.788179i \(0.288976\pi\)
\(80\) 0 0
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 13561.0 1.63760
\(92\) 0 0
\(93\) −14031.0 −1.62227
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9071.00 −0.964077 −0.482038 − 0.876150i \(-0.660103\pi\)
−0.482038 + 0.876150i \(0.660103\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −16418.0 −1.54755 −0.773777 − 0.633458i \(-0.781635\pi\)
−0.773777 + 0.633458i \(0.781635\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −18721.0 −1.57571 −0.787855 − 0.615861i \(-0.788808\pi\)
−0.787855 + 0.615861i \(0.788808\pi\)
\(110\) 0 0
\(111\) −18558.0 −1.50621
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −15471.0 −1.13018
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 14641.0 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10942.0 0.678405 0.339203 − 0.940713i \(-0.389843\pi\)
0.339203 + 0.940713i \(0.389843\pi\)
\(128\) 0 0
\(129\) 28719.0 1.72580
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 42671.0 2.41229
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −38158.0 −1.97495 −0.987475 − 0.157777i \(-0.949567\pi\)
−0.987475 + 0.157777i \(0.949567\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −23760.0 −1.09954
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −42121.0 −1.84733 −0.923666 − 0.383199i \(-0.874822\pi\)
−0.923666 + 0.383199i \(0.874822\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 12049.0 0.488823 0.244412 − 0.969672i \(-0.421405\pi\)
0.244412 + 0.969672i \(0.421405\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 51769.0 1.94847 0.974237 − 0.225527i \(-0.0724104\pi\)
0.974237 + 0.225527i \(0.0724104\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 7920.00 0.277301
\(170\) 0 0
\(171\) −48681.0 −1.66482
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −65521.0 −1.99997 −0.999985 − 0.00552484i \(-0.998241\pi\)
−0.999985 + 0.00552484i \(0.998241\pi\)
\(182\) 0 0
\(183\) −64791.0 −1.93469
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 51759.0 1.44898
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 54049.0 1.45102 0.725509 − 0.688212i \(-0.241604\pi\)
0.725509 + 0.688212i \(0.241604\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −67321.0 −1.69998 −0.849991 − 0.526797i \(-0.823393\pi\)
−0.849991 + 0.526797i \(0.823393\pi\)
\(200\) 0 0
\(201\) −79281.0 −1.96235
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 86519.0 1.94333 0.971665 − 0.236362i \(-0.0759551\pi\)
0.971665 + 0.236362i \(0.0759551\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −110689. −2.35063
\(218\) 0 0
\(219\) −76878.0 −1.60293
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 92569.0 1.86147 0.930735 − 0.365695i \(-0.119169\pi\)
0.930735 + 0.365695i \(0.119169\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 62399.0 1.18989 0.594945 − 0.803767i \(-0.297174\pi\)
0.594945 + 0.803767i \(0.297174\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −69138.0 −1.23089
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 113279. 1.95036 0.975181 − 0.221408i \(-0.0710653\pi\)
0.975181 + 0.221408i \(0.0710653\pi\)
\(242\) 0 0
\(243\) −59049.0 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 114791. 1.88154
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −146402. −2.18247
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −88318.0 −1.20257 −0.601285 − 0.799034i \(-0.705345\pi\)
−0.601285 + 0.799034i \(0.705345\pi\)
\(272\) 0 0
\(273\) −122049. −1.63760
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −12191.0 −0.158884 −0.0794419 − 0.996839i \(-0.525314\pi\)
−0.0794419 + 0.996839i \(0.525314\pi\)
\(278\) 0 0
\(279\) 126279. 1.62227
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −107111. −1.33740 −0.668700 − 0.743532i \(-0.733149\pi\)
−0.668700 + 0.743532i \(0.733149\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 83521.0 1.00000
\(290\) 0 0
\(291\) 81639.0 0.964077
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 226561. 2.50065
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 124489. 1.32085 0.660426 − 0.750891i \(-0.270376\pi\)
0.660426 + 0.750891i \(0.270376\pi\)
\(308\) 0 0
\(309\) 147762. 1.54755
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −12911.0 −0.131787 −0.0658933 − 0.997827i \(-0.520990\pi\)
−0.0658933 + 0.997827i \(0.520990\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 168489. 1.57571
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −88078.0 −0.803917 −0.401959 − 0.915658i \(-0.631670\pi\)
−0.401959 + 0.915658i \(0.631670\pi\)
\(332\) 0 0
\(333\) 167022. 1.50621
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 199249. 1.75443 0.877216 − 0.480097i \(-0.159398\pi\)
0.877216 + 0.480097i \(0.159398\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −16969.0 −0.144234
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 8402.00 0.0689814 0.0344907 − 0.999405i \(-0.489019\pi\)
0.0344907 + 0.999405i \(0.489019\pi\)
\(350\) 0 0
\(351\) 139239. 1.13018
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 230880. 1.77163
\(362\) 0 0
\(363\) −131769. −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 217849. 1.61742 0.808711 − 0.588206i \(-0.200166\pi\)
0.808711 + 0.588206i \(0.200166\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −54671.0 −0.392952 −0.196476 − 0.980509i \(-0.562950\pi\)
−0.196476 + 0.980509i \(0.562950\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 86039.0 0.598986 0.299493 − 0.954098i \(-0.403182\pi\)
0.299493 + 0.954098i \(0.403182\pi\)
\(380\) 0 0
\(381\) −98478.0 −0.678405
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −258471. −1.72580
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −313631. −1.98993 −0.994965 − 0.100219i \(-0.968046\pi\)
−0.994965 + 0.100219i \(0.968046\pi\)
\(398\) 0 0
\(399\) −384039. −2.41229
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −297769. −1.83345
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 256799. 1.53514 0.767568 − 0.640968i \(-0.221467\pi\)
0.767568 + 0.640968i \(0.221467\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 343422. 1.97495
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −226318. −1.27689 −0.638447 − 0.769666i \(-0.720423\pi\)
−0.638447 + 0.769666i \(0.720423\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −511129. −2.80333
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 121969. 0.650539 0.325270 − 0.945621i \(-0.394545\pi\)
0.325270 + 0.945621i \(0.394545\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 117239. 0.608335 0.304168 − 0.952619i \(-0.401622\pi\)
0.304168 + 0.952619i \(0.401622\pi\)
\(440\) 0 0
\(441\) 213840. 1.09954
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 379089. 1.84733
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −244898. −1.17261 −0.586304 − 0.810091i \(-0.699418\pi\)
−0.586304 + 0.810091i \(0.699418\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 152062. 0.709347 0.354673 − 0.934990i \(-0.384592\pi\)
0.354673 + 0.934990i \(0.384592\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −625439. −2.84341
\(470\) 0 0
\(471\) −108441. −0.488823
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −393842. −1.70228
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 98569.0 0.415607 0.207803 − 0.978171i \(-0.433369\pi\)
0.207803 + 0.978171i \(0.433369\pi\)
\(488\) 0 0
\(489\) −465921. −1.94847
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 226199. 0.908426 0.454213 − 0.890893i \(-0.349920\pi\)
0.454213 + 0.890893i \(0.349920\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −71280.0 −0.277301
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −606482. −2.32261
\(512\) 0 0
\(513\) 438129. 1.66482
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −97751.0 −0.357370 −0.178685 − 0.983906i \(-0.557184\pi\)
−0.178685 + 0.983906i \(0.557184\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 279841. 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −527281. −1.80156 −0.900778 − 0.434281i \(-0.857003\pi\)
−0.900778 + 0.434281i \(0.857003\pi\)
\(542\) 0 0
\(543\) 589689. 1.99997
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 342382. 1.14429 0.572145 − 0.820152i \(-0.306111\pi\)
0.572145 + 0.820152i \(0.306111\pi\)
\(548\) 0 0
\(549\) 583119. 1.93469
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −545422. −1.78354
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 609481. 1.95046
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −465831. −1.44898
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −619321. −1.89952 −0.949759 − 0.312981i \(-0.898672\pi\)
−0.949759 + 0.312981i \(0.898672\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −401231. −1.20515 −0.602577 − 0.798060i \(-0.705860\pi\)
−0.602577 + 0.798060i \(0.705860\pi\)
\(578\) 0 0
\(579\) −486441. −1.45102
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) −936959. −2.70078
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 605889. 1.69998
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 715199. 1.98006 0.990029 − 0.140863i \(-0.0449877\pi\)
0.990029 + 0.140863i \(0.0449877\pi\)
\(602\) 0 0
\(603\) 713529. 1.96235
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 74302.0 0.201662 0.100831 − 0.994904i \(-0.467850\pi\)
0.100831 + 0.994904i \(0.467850\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −516338. −1.37408 −0.687042 − 0.726618i \(-0.741091\pi\)
−0.687042 + 0.726618i \(0.741091\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 134279. 0.350451 0.175225 − 0.984528i \(-0.443935\pi\)
0.175225 + 0.984528i \(0.443935\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 793799. 1.99366 0.996832 − 0.0795399i \(-0.0253451\pi\)
0.996832 + 0.0795399i \(0.0253451\pi\)
\(632\) 0 0
\(633\) −778671. −1.94333
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −504240. −1.24268
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 728302. 1.76153 0.880764 − 0.473555i \(-0.157030\pi\)
0.880764 + 0.473555i \(0.157030\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 996201. 2.35063
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 691902. 1.60293
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −858958. −1.96593 −0.982967 − 0.183781i \(-0.941166\pi\)
−0.982967 + 0.183781i \(0.941166\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −833121. −1.86147
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −425858. −0.940231 −0.470116 − 0.882605i \(-0.655788\pi\)
−0.470116 + 0.882605i \(0.655788\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 644041. 1.39693
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −561591. −1.18989
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 782162. 1.63810 0.819050 − 0.573722i \(-0.194501\pi\)
0.819050 + 0.573722i \(0.194501\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −1.23926e6 −2.50757
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 929519. 1.84912 0.924562 − 0.381033i \(-0.124432\pi\)
0.924562 + 0.381033i \(0.124432\pi\)
\(710\) 0 0
\(711\) 622242. 1.23089
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 1.16568e6 2.24237
\(722\) 0 0
\(723\) −1.01951e6 −1.95036
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 160249. 0.303198 0.151599 − 0.988442i \(-0.451558\pi\)
0.151599 + 0.988442i \(0.451558\pi\)
\(728\) 0 0
\(729\) 531441. 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 5422.00 0.0100914 0.00504570 − 0.999987i \(-0.498394\pi\)
0.00504570 + 0.999987i \(0.498394\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 401042. 0.734346 0.367173 − 0.930153i \(-0.380326\pi\)
0.367173 + 0.930153i \(0.380326\pi\)
\(740\) 0 0
\(741\) −1.03312e6 −1.88154
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 316802. 0.561705 0.280852 − 0.959751i \(-0.409383\pi\)
0.280852 + 0.959751i \(0.409383\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 693169. 1.20962 0.604808 − 0.796371i \(-0.293250\pi\)
0.604808 + 0.796371i \(0.293250\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.32919e6 2.28317
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −732481. −1.23864 −0.619318 − 0.785140i \(-0.712591\pi\)
−0.619318 + 0.785140i \(0.712591\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.31762e6 2.18247
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 862969. 1.39330 0.696652 − 0.717409i \(-0.254672\pi\)
0.696652 + 0.717409i \(0.254672\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.37501e6 −2.18655
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 274679. 0.417622 0.208811 − 0.977956i \(-0.433041\pi\)
0.208811 + 0.977956i \(0.433041\pi\)
\(812\) 0 0
\(813\) 794862. 1.20257
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.91779e6 2.87314
\(818\) 0 0
\(819\) 1.09844e6 1.63760
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1.03769e6 1.53203 0.766015 − 0.642822i \(-0.222237\pi\)
0.766015 + 0.642822i \(0.222237\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1.16472e6 −1.69477 −0.847387 − 0.530976i \(-0.821825\pi\)
−0.847387 + 0.530976i \(0.821825\pi\)
\(830\) 0 0
\(831\) 109719. 0.158884
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.13651e6 −1.62227
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 707281. 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.03951e6 −1.44898
\(848\) 0 0
\(849\) 963999. 1.33740
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 69889.0 0.0960530 0.0480265 − 0.998846i \(-0.484707\pi\)
0.0480265 + 0.998846i \(0.484707\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 534962. 0.724998 0.362499 − 0.931984i \(-0.381924\pi\)
0.362499 + 0.931984i \(0.381924\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −751689. −1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −1.68252e6 −2.21781
\(872\) 0 0
\(873\) −734751. −0.964077
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.44427e6 −1.87780 −0.938900 − 0.344189i \(-0.888154\pi\)
−0.938900 + 0.344189i \(0.888154\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1.36313e6 1.74830 0.874149 − 0.485658i \(-0.161420\pi\)
0.874149 + 0.485658i \(0.161420\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −776882. −0.982996
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −2.03905e6 −2.50065
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.59950e6 1.94433 0.972166 − 0.234295i \(-0.0752782\pi\)
0.972166 + 0.234295i \(0.0752782\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −746281. −0.883632 −0.441816 − 0.897106i \(-0.645666\pi\)
−0.441816 + 0.897106i \(0.645666\pi\)
\(920\) 0 0
\(921\) −1.12040e6 −1.32085
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1.32986e6 −1.54755
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −1.58664e6 −1.83054
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.65547e6 −1.88557 −0.942784 − 0.333403i \(-0.891803\pi\)
−0.942784 + 0.333403i \(0.891803\pi\)
\(938\) 0 0
\(939\) 116199. 0.131787
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −1.63152e6 −1.81159
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.50696e6 1.63175
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −482978. −0.516505 −0.258252 − 0.966077i \(-0.583147\pi\)
−0.258252 + 0.966077i \(0.583147\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 2.70922e6 2.86166
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.51640e6 −1.57571
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.05996e6 1.07930 0.539649 − 0.841890i \(-0.318557\pi\)
0.539649 + 0.841890i \(0.318557\pi\)
\(992\) 0 0
\(993\) 792702. 0.803917
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.59922e6 −1.60886 −0.804428 − 0.594050i \(-0.797528\pi\)
−0.804428 + 0.594050i \(0.797528\pi\)
\(998\) 0 0
\(999\) −1.50320e6 −1.50621
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 300.5.g.a.101.1 ✓ 1
3.2 odd 2 CM 300.5.g.a.101.1 ✓ 1
5.2 odd 4 300.5.b.b.149.2 2
5.3 odd 4 300.5.b.b.149.1 2
5.4 even 2 300.5.g.c.101.1 yes 1
15.2 even 4 300.5.b.b.149.2 2
15.8 even 4 300.5.b.b.149.1 2
15.14 odd 2 300.5.g.c.101.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.5.b.b.149.1 2 5.3 odd 4
300.5.b.b.149.1 2 15.8 even 4
300.5.b.b.149.2 2 5.2 odd 4
300.5.b.b.149.2 2 15.2 even 4
300.5.g.a.101.1 ✓ 1 1.1 even 1 trivial
300.5.g.a.101.1 ✓ 1 3.2 odd 2 CM
300.5.g.c.101.1 yes 1 5.4 even 2
300.5.g.c.101.1 yes 1 15.14 odd 2